Garlic definitely kills parasites. Here are the references I used for the article I wrote that was published in Aquarium Fish International Magazine this April:
Ankri, S., and D. Mirelman. 2001. “Antimicrobial properties of allicin from garlic.” Microbes and Infection 2:125-129
Boxaspen, K., and J.C. Holm. 1992. “New biocides used against sea lice compared to organo-phosphorous compounds.” European Aquaculture Society
Coppi, A., M. Cabinian, D. Mirelman, and P. Sinnis. 2006. “Antimalarial Activity of Allicin, a Biologically Active Compound from Garlic Cloves.” Antimicrobial Agents and Chemotherapy 5:1731-1737
D.L.N. 1998. “Allicin, one of the active principles of garlic, inhibits the growth of protozoan parasites.” Chemtech 4:45
Holden, C. 1997. “Fighting Parasites With Garlic.” Science. 5338:581
Soko, C. K., and D.E. Barker. 2004. “Efficacy of crushed garlic and lemon juice as bio-product treatments for Ichthyophthirius multifiliis (‘ich’) infections among juvenile Nile tilapia, Oreochromis niloticus.” Aquaculture Association of Canada 9:108-110